# Torsion

• Oct 24th 2009, 07:29 AM
Thomas154321
Torsion
I'm looking to derive the equation for torsion for a space curve $\displaystyle \alpha (u)$

$\displaystyle \tau= \frac{(\alpha ' \times \alpha '')\cdot \alpha '''}{{||\alpha '\times \alpha ''||}^{2}}$

I think I use $\displaystyle b = t \times n$ and of course $\displaystyle t = \frac{\alpha '}{||\alpha '||}$

Then $\displaystyle \frac{db}{ds}=\frac{1}{||\alpha '||}\cdot\frac{db}{du}=\tau n$

But I can't see where the $\displaystyle \alpha '''$ comes in...
• Oct 24th 2009, 08:00 AM
Rebesques
Quote:

But I can't see where the $\displaystyle \alpha '''$ comes in...

From $\displaystyle \frac{db}{ds}$.
• Oct 24th 2009, 08:23 AM
Thomas154321
First I need $\displaystyle \frac{db}{du}=\frac{d}{du}(t \times n) = t' \times n + t \times n'$

Now $\displaystyle n = \frac{t'}{||t'||}$.

How do I find t' and n' ?
• Oct 24th 2009, 10:09 AM
Rebesques
Well... First of all, you can reparametrize for arc length, so that $\displaystyle ||\alpha||=1$, $\displaystyle t=\alpha'$, $\displaystyle n=\alpha''/||\alpha''||$ and eventually $\displaystyle b'=\left(\alpha'\times\frac{\alpha''}{||\alpha''|| }\right)'$, etc...
• Oct 25th 2009, 08:43 AM
Thomas154321
What if the aim is to find the torsion for any parametrisation?